# 1.3 Radicals and rational expressions  (Page 5/11)

 Page 5 / 11

## Simplifying rational exponents

Simplify:

1. $5\left(2{x}^{\frac{3}{4}}\right)\left(3{x}^{\frac{1}{5}}\right)$
2. ${\left(\frac{16}{9}\right)}^{-\frac{1}{2}}$

Simplify $\text{\hspace{0.17em}}{\left(8x\right)}^{\frac{1}{3}}\left(14{x}^{\frac{6}{5}}\right).$

$28{x}^{\frac{23}{15}}$

Access these online resources for additional instruction and practice with radicals and rational exponents.

## Key concepts

• The principal square root of a number $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is the nonnegative number that when multiplied by itself equals $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ See [link] .
• If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ are nonnegative, the square root of the product $\text{\hspace{0.17em}}ab\text{\hspace{0.17em}}$ is equal to the product of the square roots of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ See [link] and [link] .
• If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ are nonnegative, the square root of the quotient $\text{\hspace{0.17em}}\frac{a}{b}\text{\hspace{0.17em}}$ is equal to the quotient of the square roots of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ See [link] and [link] .
• Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See [link] and [link] .
• The principal n th root of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is the number with the same sign as $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ that when raised to the n th power equals $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ These roots have the same properties as square roots. See [link] .
• The properties of exponents apply to rational exponents. See [link] .

## Verbal

What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.

When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.

Where would radicals come in the order of operations? Explain why.

Every number will have two square roots. What is the principal square root?

The principal square root is the nonnegative root of the number.

Can a radical with a negative radicand have a real square root? Why or why not?

## Numeric

For the following exercises, simplify each expression.

$\sqrt{256}$

16

$\sqrt{\sqrt{256}}$

$\sqrt{4\left(9+16\right)}$

10

$\sqrt{289}-\sqrt{121}$

$\sqrt{196}$

14

$\sqrt{1}$

$\sqrt{98}$

$7\sqrt{2}$

$\sqrt{\frac{27}{64}}$

$\sqrt{\frac{81}{5}}$

$\frac{9\sqrt{5}}{5}$

$\sqrt{800}$

$\sqrt{169}+\sqrt{144}$

25

$\sqrt{\frac{8}{50}}$

$\frac{18}{\sqrt{162}}$

$\sqrt{2}$

$\sqrt{192}$

$14\sqrt{6}-6\sqrt{24}$

$2\sqrt{6}$

$15\sqrt{5}+7\sqrt{45}$

$\sqrt{150}$

$5\sqrt{6}$

$\sqrt{\frac{96}{100}}$

$\left(\sqrt{42}\right)\left(\sqrt{30}\right)$

$6\sqrt{35}$

$12\sqrt{3}-4\sqrt{75}$

$\sqrt{\frac{4}{225}}$

$\frac{2}{15}$

$\sqrt{\frac{405}{324}}$

$\sqrt{\frac{360}{361}}$

$\frac{6\sqrt{10}}{19}$

$\frac{5}{1+\sqrt{3}}$

$\frac{8}{1-\sqrt{17}}$

$-\frac{1+\sqrt{17}}{2}$

$\sqrt[4]{16}$

$\sqrt[3]{128}+3\sqrt[3]{2}$

$7\sqrt[3]{2}$

$\sqrt[5]{\frac{-32}{243}}$

$\frac{15\sqrt[4]{125}}{\sqrt[4]{5}}$

$15\sqrt{5}$

$3\sqrt[3]{-432}+\sqrt[3]{16}$

## Algebraic

For the following exercises, simplify each expression.

$\sqrt{400{x}^{4}}$

$20{x}^{2}$

$\sqrt{4{y}^{2}}$

$\sqrt{49p}$

$7\sqrt{p}$

${\left(144{p}^{2}{q}^{6}\right)}^{\frac{1}{2}}$

${m}^{\frac{5}{2}}\sqrt{289}$

$18{m}^{2}\sqrt{m}$

$9\sqrt{3{m}^{2}}+\sqrt{27}$

$3\sqrt{a{b}^{2}}-b\sqrt{a}$

$2b\sqrt{a}$

$\frac{4\sqrt{2n}}{\sqrt{16{n}^{4}}}$

$\sqrt{\frac{225{x}^{3}}{49x}}$

$\frac{15x}{7}$

$3\sqrt{44z}+\sqrt{99z}$

$\sqrt{50{y}^{8}}$

$5{y}^{4}\sqrt{2}$

$\sqrt{490b{c}^{2}}$

$\sqrt{\frac{32}{14d}}$

$\frac{4\sqrt{7d}}{7d}$

${q}^{\frac{3}{2}}\sqrt{63p}$

$\frac{\sqrt{8}}{1-\sqrt{3x}}$

$\frac{2\sqrt{2}+2\sqrt{6x}}{1-3x}$

$\sqrt{\frac{20}{121{d}^{4}}}$

${w}^{\frac{3}{2}}\sqrt{32}-{w}^{\frac{3}{2}}\sqrt{50}$

$-w\sqrt{2w}$

$\sqrt{108{x}^{4}}+\sqrt{27{x}^{4}}$

$\frac{\sqrt{12x}}{2+2\sqrt{3}}$

$\frac{3\sqrt{x}-\sqrt{3x}}{2}$

$\sqrt{147{k}^{3}}$

$\sqrt{125{n}^{10}}$

$5{n}^{5}\sqrt{5}$

$\sqrt{\frac{42q}{36{q}^{3}}}$

$\sqrt{\frac{81m}{361{m}^{2}}}$

$\frac{9\sqrt{m}}{19m}$

$\sqrt{72c}-2\sqrt{2c}$

$\sqrt{\frac{144}{324{d}^{2}}}$

$\frac{2}{3d}$

$\sqrt[3]{24{x}^{6}}+\sqrt[3]{81{x}^{6}}$

$\sqrt[4]{\frac{162{x}^{6}}{16{x}^{4}}}$

$\frac{3\sqrt[4]{2{x}^{2}}}{2}$

$\sqrt[3]{64y}$

$\sqrt[3]{128{z}^{3}}-\sqrt[3]{-16{z}^{3}}$

$6z\sqrt[3]{2}$

$\sqrt[5]{1,024{c}^{10}}$

## Real-world applications

A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating $\text{\hspace{0.17em}}\sqrt{90,000+160,000}.\text{\hspace{0.17em}}$ What is the length of the guy wire?

500 feet

A car accelerates at a rate of where t is the time in seconds after the car moves from rest. Simplify the expression.

## Extensions

For the following exercises, simplify each expression.

$\frac{\sqrt{8}-\sqrt{16}}{4-\sqrt{2}}-{2}^{\frac{1}{2}}$

$\frac{-5\sqrt{2}-6}{7}$

$\frac{{4}^{\frac{3}{2}}-{16}^{\frac{3}{2}}}{{8}^{\frac{1}{3}}}$

$\frac{\sqrt{m{n}^{3}}}{{a}^{2}\sqrt{{c}^{-3}}}\cdot \frac{{a}^{-7}{n}^{-2}}{\sqrt{{m}^{2}{c}^{4}}}$

$\frac{\sqrt{mnc}}{{a}^{9}cmn}$

$\frac{a}{a-\sqrt{c}}$

$\frac{x\sqrt{64y}+4\sqrt{y}}{\sqrt{128y}}$

$\frac{2\sqrt{2}x+\sqrt{2}}{4}$

$\left(\frac{\sqrt{250{x}^{2}}}{\sqrt{100{b}^{3}}}\right)\left(\frac{7\sqrt{b}}{\sqrt{125x}}\right)$

$\sqrt{\frac{\sqrt[3]{64}+\sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}$

$\frac{\sqrt{3}}{3}$

12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8